158 VIII DIFFERENTIAL CALCULUS
(b) Suppose E has finite dimension. Show that if/is quasi-differentiable at XQ e A,
/is differentiable at x0. (Use contradiction: let u be the quasi-derivative of/at J*r0,
and suppose there is a > 0 and a sequence (xn) of points of A, tending to XQ , such that
!!/(**) — /(*o) — u* (xn — Xo)\\^ oc\\xn — jcoll- Using the local compactness of E,
show that one may suppose that the sequence (\\x» — x0\\) is decreasing, and that the
sequence of the vectors zn — (xn — xo)/\\xn — XQ\\ tends to a limit in E; then define a
continuous mapping g of [0,1] into E such that ^(0) — ^0» that g'(ft) exists, but that
«(#'(0)) is not the derivative of t -+f(g(t)) at / = 0.)
6. Let 1= [0,1], and let E be the Banach space ^R(I). In order that the mapping
x-> [Ml of E into R be quasi-differentiable at a point *0, it is necessary and sufficient
that the function t-+[xo(t)\ reaches its maximum in I at a single point to el; the
quasi-derivative of x -> \\x\\ at x0 is then the linear mapping u such that u(z) ~ z(tQ) if
x0(/o) > 0, u(z) = -z(/0) if *o(A>) < 0 (compare Section 8.2, Problem 3). (To prove the
condition is necessary, suppose \XQ\ reaches its maximum at two distinct points t0, h at
least; let y be a continuous mapping of I into itself, equal to 1 at t0, to 0 at tj.; examine
the behavior of (\\XQ -\-Xy\\— \\x0 JQ/A as the real number A ^ 0 tends to 0. To prove the
condition is sufficient, let A -*• ZA be a continuous mapping of I into E, having a derivative
a e E at A = 0 and such that z0 =0; observe that if tj, is the largest number in I (or the
smallest number in I) where t -+ [*0(0 + z*(f)l reaches its maximum, then *A tends to t0
when A tends to 0.) Deduce from that result that the mapping x -» ||jt || of E into R is not
differentiable at any point (compare to Section 8.2, Problem 2).
7. Let/be a continuous mapping of an open subset A of a Banach space E into a Banach
space F. Suppose/is lipschitzian in A: this means (7.6, Problem 12) that there exists a
constant k > 0 such that \\f(xt) -f(x2) II ^ k ||*i - x2 \\ for any pair of points of A. Let
XQ e A, and suppose there is a linear mapping u of E into F such that, for any vector
a *£ 0 in E, the limit of (/(x0 + at) —f(.x0))/t when / ^ 0 tends to 0 in R, exists and is
equal to u(a). Show that/is quasi-differentiable at x0.
8. (a) Let a, b be two points in a Banach space E. Show that the mapping t -> ||<2 + tb \\ of
R into itself has a derivative on the right and a derivative on the left for every / e R
(prove that if 0<t<sy then (\\a+bt\\- \\a\Q/t^(\\a + bs\\— \\a\\)/s and use (4.2.1)).
(b) Let u be a continuous mapping of an interval I e R into E. Show that if at a point
t0 E I, u has a derivative on the right, then f-> \\u(t) || has at t0 a derivative on the right
and
(D+JMIX'o) ^ ]|D+«('o)ll
(apply (a)).
(c) Let U be a continuous mapping of I into &(E; E). Show that if at a point t0 e I,
U has a derivative on the right and U(tQ) is a linear homeomorphism of E onto itself,
then the mapping t-*\\(U(t)')-1\\ =/(0, which is defined in a neighborhood of /0»
has a derivative on the right at /0» and that
5. THE MEAN VALUE THEOREM
(8.5.1) Let I = [a, j8] be a compact interval in R, / a continuous mapping
of Unto a Banach space F, cp a continuous mapping of I into R. We suppose
that there is a denumerable subset D such that, for each { e I — D, / and q>
have both a derivative at £ with respect to I (8.4), and that